The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state asTheorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by1then2unless f≡0. The constant is the best possible.
Z. Opial [11] proved in 1960 the following theorem:Theorem 1. If u is a continuously differentiable function on [0, b], and if u(0)= u(b)=0 and u(x) > 0 for x ∊ (0, b), then1where the constant b/4 is the best possible.
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